Explicit expression of stationary response probability density for nonlinear stochastic systems

2021 ◽  
Author(s):  
Xiaoling Jin ◽  
Yanping Tian ◽  
Yong Wang ◽  
Zhilong Huang
Keyword(s):  
Probability Density ◽  
Explicit Expression ◽  
Stochastic Systems ◽  
Response Probability ◽  
Nonlinear Stochastic Systems ◽  
Stationary Response

scholarly journals Stationary Response of a Class of Nonlinear Stochastic Systems Undergoing Markovian Jumps

2015 ◽  
Vol 82 (5) ◽  
Author(s):  
Rong-Hua Huan ◽  
Wei-qiu Zhu ◽  
Fai Ma ◽  
Zu-guang Ying
Keyword(s):  
Stochastic Systems ◽  
Original System ◽  
Stochastic Averaging ◽  
Kolmogorov Equation ◽  
Markovian Jump ◽  
Nonlinear Stochastic Systems ◽  
Stationary Response ◽  
Set Up ◽  
Jump Systems ◽  
Stationary Probabilities

Systems whose specifications change abruptly and statistically, referred to as Markovian-jump systems, are considered in this paper. An approximate method is presented to assess the stationary response of multidegree, nonlinear, Markovian-jump, quasi-nonintegrable Hamiltonian systems subjected to stochastic excitation. Using stochastic averaging, the quasi-nonintegrable Hamiltonian equations are first reduced to a one-dimensional Itô equation governing the energy envelope. The associated Fokker–Planck–Kolmogorov equation is then set up, from which approximate stationary probabilities of the original system are obtained for different jump rules. The validity of this technique is demonstrated by using a nonlinear two-degree oscillator that is stochastically driven and capable of Markovian jumps.

Reshaping of the probability density function of nonlinear stochastic systems against abrupt changes

2019 ◽  
Vol 26 (7-8) ◽  
pp. 532-539
Author(s):  
Lei Xia ◽  
Ronghua Huan ◽  
Weiqiu Zhu ◽  
Chenxuan Zhu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Stochastic Systems ◽  
Stationary Probability ◽  
Markov Jump ◽  
Nonlinear Stochastic Systems ◽  
Stationary Probability Density ◽  
Abrupt Changes ◽  
Stationary Probability Density Function

The operation of dynamic systems is often accompanied by abrupt and random changes in their configurations, which will dramatically change the stationary probability density function of their response. In this article, an effective procedure is proposed to reshape the stationary probability density function of nonlinear stochastic systems against abrupt changes. Based on the Markov jump theory, such a system is formulated as a continuous system with discrete Markov jump parameters. The limiting averaging principle is then applied to suppress the rapidly varying Markov jump process to generate a probability-weighted system. Then, the approximate expression of the stationary probability density function of the system is obtained, based on which the reshaping control law can be designed, which has two parts: (i) the first part (conservative part) is designed to make the reshaped system and the undisturbed system have the same Hamiltonian; (ii) the second (dissipative part) is designed so that the stationary probability density function of the reshaped system is the same as that of undisturbed system. The proposed law is exactly analytical and no online measurement is required. The application and effectiveness of the proposed procedure are demonstrated by using an example of three degrees-of-freedom nonlinear stochastic system subjected to abrupt changes.

Stationary response probability density of nonlinear random vibrating systems: a data-driven method

2020 ◽  
Vol 100 (3) ◽  
pp. 2337-2352
Author(s):  
Yanping Tian ◽  
Yong Wang ◽  
Hanqing Jiang ◽  
Zhilong Huang ◽  
Isaac Elishakoff ◽  
...  
Keyword(s):  
Probability Density ◽  
Response Probability ◽  
Data Driven ◽  
Stationary Response ◽  
Vibrating Systems
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